Projective Geometry

PROJECTIVE PROJECTIVE GEOMETRYGEOMETRY

Projective GeometryProjective Geometry

enables a clearer understanding of some more enables a clearer understanding of some more generic properties of geometric objects. generic properties of geometric objects.

is a is a non-Euclidean geometrynon-Euclidean geometry that formalizes one that formalizes one of the central principles of perspective art: that of the central principles of perspective art: that parallelparallel lines meet at lines meet at infinityinfinity and therefore are to and therefore are to be drawn that way. be drawn that way.

HistoryHistoryearly Italian early Italian RenaissanceRenaissance

architectural drawings:architectural drawings: FilippoFilippo Brunelleschi Brunelleschi (1377–1446) (1377–1446) Leon Battista Leon Battista AlbertiAlberti (1404–1472) (1404–1472)

invented the method of invented the method of perspectiveperspective drawing. drawing.

HistoryHistory

Definition of Terms:Definition of Terms:PlanePlane – the imaginary flat surface with – the imaginary flat surface with

length and width but no thickness, it may length and width but no thickness, it may extend indefinitely in any direction.extend indefinitely in any direction.

Station point (SP)Station point (SP) – the point where the – the point where the observer is supposed to stand while observer is supposed to stand while viewing the object.viewing the object.

Picture plane (PP)Picture plane (PP) – the plane in which the – the plane in which the object is located.object is located.

HorizonHorizon – the horizontal plane at level.– the horizontal plane at level.

Vanishing point (VP)Vanishing point (VP) – a point wherein the – a point wherein the edges of the object will imaginarily meet.edges of the object will imaginarily meet.

Vanishing line (VL)Vanishing line (VL) – the rays from the – the rays from the vanishing point from the edges which vanishing point from the edges which serves as the guide in locating the ends of serves as the guide in locating the ends of the object.the object.

Ground plane (GP)Ground plane (GP) – a plane which is in-– a plane which is in-line with the eye.line with the eye.

Ground line (GL)Ground line (GL) – the intersection of the – the intersection of the ground plane and the picture plane.ground plane and the picture plane.

Basic elements:

Points LinesPlanes

Basic elements:

Points

Basic elements:

Points

Basic elements:

PointsLines

Basic elements:

PointsLinesPlanes

Geometric TransformationsGeometric Transformations Reflections - Reflections - is a type of transformation where the line is a type of transformation where the line

of symmetry is a of symmetry is a perpendicular bisectorperpendicular bisector of  of corresponding points. corresponding points.

Translations - Translations - preserve preserve congruencycongruency. This means the . This means the image and pre-image of an object is image and pre-image of an object is exactly the same exactly the same shape and sizeshape and size as the pre-image . as the pre-image .

Rotation - Rotation - moves all the points of the figure moves all the points of the figure the the same same angleangle around the same pivot point. around the same pivot point.

Dilations - Dilations - is a scaled transformation. The distance is a scaled transformation. The distance from each point in the pre-image to the center is from each point in the pre-image to the center is multiplied by the scale factor to find the distance multiplied by the scale factor to find the distance along in the same direction to the corresponding image along in the same direction to the corresponding image point.point.

Reflections

ReflectionsReflections The pre-image ABC is reflected onto the image The pre-image ABC is reflected onto the image

A'B'C' across the A'B'C' across the x-axisx-axis..

Reflections

Translations

TranslationsTranslations The pre-image is translated onto the image The pre-image is translated onto the image 4 4

units rightunits right..

Rotations

RotationsRotations The pre-image ABC is rotated to image A'B'C' The pre-image ABC is rotated to image A'B'C'

about about P(0,0)P(0,0) as the as the center of rotationcenter of rotation and and through through 90°90° as the as the angle of rotationangle of rotation. .

Dilations

DilationsDilations The triangle is transformed by a dilation with the The triangle is transformed by a dilation with the

center at (0,0) center at (0,0) and a and a scale factor of 2scale factor of 2..

Axonometric ProjectionAxonometric Projection the ability to show the inclined position of an the ability to show the inclined position of an

object with respect to the plane of projection.object with respect to the plane of projection.

Kinds:Kinds: IsometricIsometric – – IsoIso (one or equal) (one or equal) and and Metrus Metrus

(measures(measures); equal measures.); equal measures. DimetricDimetric – an axonometric drawing into – an axonometric drawing into twotwo angle. angle. TrimetricTrimetric – utilizes – utilizes threethree different angles. different angles.

Axonometric ProjectionAxonometric Projection

Isometric Projection

In isometric projection the angles between the projection of the axes are equal i.e. 120º.

It is important to appreciate that it is the angles between the projection of the axes that are being discussed and not the true angles between the axes themselves which is always 90º.

Isometric Projection

Isometric Projection

Isometric Projection

Isometric Projection

Dimetric Projection

The angles between the projection of the axes in The angles between the projection of the axes in dimetric projection renders two of the three to be dimetric projection renders two of the three to be equal. equal.

To draw the outline of an object in dimetric To draw the outline of an object in dimetric projection, two scales are required. projection, two scales are required.

The scales are generated the same as for The scales are generated the same as for isometricisometric. .

Dimetric Projection

Dimetric Projection

Trimetric Projection

In trimetric projection the projection of the three angles between the axes are unequal.

Thus, three separate scales are needed to generate a trimetric projection of an object.

The scales are constructed using the same method described in isometric and dimetric projection.

Trimetric Projection

Trimetric Projection

Orthographic ProjectionOrthographic Projection

Orthographic projection shows complex objects Orthographic projection shows complex objects by doing a 2D drawing of each side to show the by doing a 2D drawing of each side to show the main features. main features.

Orthographic drawings usually consist of a front Orthographic drawings usually consist of a front view, a side view and a top view, but more views view, a side view and a top view, but more views may be shown for complex objects with lots of may be shown for complex objects with lots of detail. detail.

Here are three orthographic views of an object. Here are three orthographic views of an object.

Orthographic Projection

Orthographic Projection

Orthographic Projection

Perspective ProjectionPerspective Projection

from Latin from Latin perspicereperspicere which means “to see which means “to see clearly”.clearly”.

is the most attractive type of presenting an is the most attractive type of presenting an object, the subject appears as it seen by the object, the subject appears as it seen by the naked eye.naked eye.

It is a photographic or “picture like” result.It is a photographic or “picture like” result.

Perspective Projection

One Point Perspective One Point Perspective

Also called Also called parallel perspectiveparallel perspective..Occurs when one of its faces is parallel to Occurs when one of its faces is parallel to

the plane of projection.the plane of projection.Has only one vanishing point used.Has only one vanishing point used.

One Point PerspectiveOne Point Perspective

Even though change in eye position or tilt of Even though change in eye position or tilt of head affects the vanishing point, the view is head affects the vanishing point, the view is still one point perspective.still one point perspective.

Even though change in eye position or tilt of head affects the vanishing point, the view is still one point perspective.

Two Point Perspective Two Point Perspective

Also called Also called angular perspectiveangular perspective.. If it employs two vanishing points and the If it employs two vanishing points and the

sides are angular with the picture plane.sides are angular with the picture plane.

Two Point Perspective Two Point Perspective

Three Point Perspective Three Point Perspective

In In oblique perspectiveoblique perspective, three vanishing , three vanishing points are employed.points are employed.

If the projection plane is not parallel to any If the projection plane is not parallel to any principal axis, a three-point projection principal axis, a three-point projection occurs. occurs.

Only the edges are perpendicular with the Only the edges are perpendicular with the plane of projection and will show its true plane of projection and will show its true dimension.dimension.

Three Point Perspective

THE ENDTHE END